Minimax rates for statistical inverse problems under general source conditions

We describe the minimax reconstruction rates in linear ill-posed equations in Hilbert space when smoothness is given in terms of general source sets. The underlying fundamental result, the minimax rate on ellipsoids, is proved similarly to the seminal study by D. L. Donoho, R.~C. Liu, and B. MacGibb...

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Bibliographic Details
Published inarXiv.org
Main Authors Ding, LiTao, Mathé, Peter
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 15.11.2017
Subjects
Ellipsoids
Hilbert space
Ill posed problems
Inverse problems
Minimax technique
Smoothness
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Summary:We describe the minimax reconstruction rates in linear ill-posed equations in Hilbert space when smoothness is given in terms of general source sets. The underlying fundamental result, the minimax rate on ellipsoids, is proved similarly to the seminal study by D. L. Donoho, R.~C. Liu, and B. MacGibbon, {\it Minimax risk over hyperrectangles, and implications}, Ann.~ Statist. 18, 1990. These authors highlighted the special role of the truncated series estimator, and for such estimators the risk can explicitly be given. We provide several examples, indicating results for statistical estimation in ill-posed problems in Hilbert space.
ISSN:2331-8422